Question: Simplify and expand the following expression: $ \dfrac{9}{4p + 3}-\dfrac{p + 8}{p - 3} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(4p + 3)(p - 3)$ Multiply the first term by $\dfrac{p - 3}{p - 3}$ $ \begin{align*} \dfrac{9}{4p + 3} \times \dfrac{p - 3}{p - 3} & = \dfrac{(9)(p - 3)}{(4p + 3)(p - 3)} \\ & = \dfrac{9p - 27}{(4p + 3)(p - 3)}\end{align*} $ Multiply the second term by $\dfrac{4p + 3}{4p + 3}$ $ \begin{align*} \dfrac{p + 8}{p - 3} \times \dfrac{4p + 3}{4p + 3} & = \dfrac{(p + 8)(4p + 3)}{(p - 3)(4p + 3)} \\ & = \dfrac{4p^2 + 35p + 24}{(p - 3)(4p + 3)}\end{align*} $ Now we have: $ = \dfrac{9p - 27}{(4p + 3)(p - 3)} - \dfrac{4p^2 + 35p + 24}{(p - 3)(4p + 3)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{9p - 27 - (4p^2 + 35p + 24)}{(4p + 3)(p - 3)} $ $ = \dfrac{9p - 27 - 4p^2 - 35p - 24}{(4p + 3)(p - 3)} $ $ = \dfrac{-26p - 51 - 4p^2}{(4p + 3)(p - 3)}$ Expand the denominator: $ = \dfrac{-26p - 51 - 4p^2}{4p^2 - 9p - 9}$